The formalism developed in [1]} allows, in
principle, to treat both the continuous case of magnetic Schrödinger
operators defined on \(L^2\big (\mathbb {R}^2\big )\)  and the discrete case of tight-binding
magnetic Hamiltonians defined on \(\ell ^2\big (\mathbb {Z}^2\big )\) . However, many of the crucial
results contained in [1]}, like the
first and second Connes' formulas [1]} are proved only for the discrete
case. The extension to the continuous case of these results is not simply a
matter of a straightforward generalization but requires the change of some
structural elements for the construction of the appropriate NGC. We will try to
clarify these points below, but first, it is worth making a methodological remark:
in order to focus on the crucial points of the discussion, we will consider in
this work only the purely magnetic aspects of the dynamics, avoiding the
technical complications associated with the presence of disorder usually present in the matter. In fact,
once the magnetic aspects of the problem have been understood, the effects due
to the (random) electrostatic interaction with matter can be restored in the theory in a standard,
but not necessarily trivial, way. The general random case will be the subject
of a future work.
